What the average leaves out · Epistemic Physics

Suppose two scientific committees vote on the trajectory of a spacecraft.

Committee A splits completely East versus West: half the members vote East, half vote West.

Committee B splits completely North versus South: half vote North, half vote South.

Both committees produce the exact same mean velocity: zero! Yet their internal disagreements are entirely orthogonal.

A single average point discards this crucial geometry. To recover it, we must look at higher moments.

Covariance and the second moment

In statistics, the mean is the first moment. The second moment captures how the coordinates fluctuate together around that mean.

Under a directional Dirichlet process with base distribution $H$ and total concentration $τ_{E}$ , the covariance matrix of the velocity resultant $V$ is:

The matrix $n n^{T}$ collects outer products of the directional unit vectors.

Covariance recovers the axis along which the disagreement took place.

Cov (V) = \frac{E_{H} [n n^{T}] - v v^{T}}{τ_{E} + 1}

Equal centers, different uncertainty ellipses

Consider our two committees under a uniform circle prior with unit concentration.

One East and one West observation give a horizontal covariance ellipse: $Cov = diag (5 / 24, 1 / 24)$ .

One North and one South observation give a vertical covariance ellipse: $Cov = diag (1 / 24, 5 / 24)$ .

Both states have identical total concentration ( $τ_{E} = 3$ ), identical cone norm ( $m_{E} = 3$ ), and identical zero mean velocity. But their uncertainty ellipses point in opposite directions!

Your turn

Rotate the two opposing observations around the circle in the lab. Watch the covariance ellipse turn in real time while the mean point and cone norm remain pinned at the center.

Interactive · 03

Same center. Different story.

Two equally weighted trials point in opposite directions. Their average is always zero. Turn the rose pair by dragging either endpoint and watch its ellipse turn. The dashed blue east-and-west pair stays visible for comparison on the same axes.

Orientation 90°

Total concentration τ_E: 3
Mean velocity: (0, 0)
Cone norm m_E: 3

Same mean and amount; compare the spread
Variance	East & west	Rotated 90°
In x	0.2083	0.0417
In y	0.0417	0.2083

East & west referenceRotated pair: 90°

Each ellipse shows principal standard deviations of the Dirichlet-process resultant, with a uniform circle prior of total one. Their sizes match; only their orientations differ. These are covariance visualizations, not probability contours. Focus the rose pair and use arrow keys to rotate it; Shift turns by 10°.

Where the extra shape could help

Imagine steering a spacecraft with an uncertain starting direction. A horizontal cloud of possible directions and a vertical cloud can spread into different futures, even when their mean agrees.

This suggests an experiment: propagate the full directional belief alongside its mean, then compare which description anticipates the spread of later paths. In a sensitive dynamical system, small initial differences can grow quickly. Covariance gives us a first way to retain those differences and ask whether they help prediction.

The two committees already show the exact mathematical point: matching the mean and cone norm leaves room for different shapes of disagreement.